Two exponential-type integrators for the “good” Boussinesq equation

Abstract

We introduce two exponential-type integrators for the "good" Bousinessq equation. They are of orders one and two, respectively, and they require lower regularity of the solution compared to the classical exponential integrators. More precisely, we will prove first-order convergence in H r for solutions in H r+1 with r > 1/2 for the derived first-order scheme. For the second integrator, we prove second-order convergence in H r for solutions in H r+3 with r > 1/2 and convergence in L 2 for solutions in H 3 . Numerical results are reported to illustrate the established error estimates. The experiments clearly demonstrate that the new exponential-type integrators are favorable over classical exponential integrators for initial data with low regularity. IntroductionConsider the "good" Boussinesq (GB) equation [4] z tt + z xxxx − z xx − (z 2 ) xx = 0, (1.1) which was originally introduced as a model for one-dimensional weakly nonlinear dispersive waves in shallow water. Similar to the well-known Korteweg-de Vries (KdV) equation and the cubic Schrödinger equation, the GB equation is one of the important models describing the interaction between nonlinearity and dispersion. The GB equation has been widely applied in many areas, e.g., plasma, coastal engineering, hydraulics studies, elastic crystals and so on.The GB equation and its various extensions have been extensively analyzed in the literature. For the well-posedness, we refer to [10-12, 17, 23, 26] and references therein. For the interaction of solitary waves, we refer to [21,22]. Many numerical methods have been developed for solving the GB equation, such as finite difference methods (FDM) [5,24], Petrov-Galerkin methods [21], mesh free methods [8], Fourier spectral methods [6,7,27] and operator splitting methods [28,29]. Regarding the numerical analysis for the GB equation,

Error analysis of a class of semi-discrete schemes for solving the Gross-Pitaevskii equation at low regularity

“…The goal of this section is to construct a first order discretization of the oscillatory integral (14) when working on a general domain Ω, and which allows for the improved local error structure (15) established in the preceding section. This is achieved by introducing a properly chosen filtered function which will filter out the dominant oscillatory terms L dom,1 , L dom,2 explicitely found in the preceding section.…”

“…Hence, we recover the discretization of the oscillatory integral (14) together with an improved local error structure of the form (15);…”

“…Recently much progress has been made in the development of low-regularity approximations to nonlinear evolution equations. First, in the case of periodic boundary conditions a class of schemes called Fourier integrators [12] or resonance based schemes [4] were introduced to approximate the time dynamics of dispersive equations such as NLS, KdV, Boussinesq, Dirac and Klein-Gordon (see [8], [11], [14] , [17], [5]). Very recently higher order extensions of these resonance based schemes were introduced in [4] for approximating in a unified fashion a large class of dispersive equations with periodic boundary conditions.…”

Resonance-based schemes for dispersive equations via decorated trees

“…While first-order resonance-based discretisations have been presented for particular examples – for example, the Nonlinear Schrödinger (NLS), Korteweg–de Vries (KdV), Boussinesq, Dirac and Klein–Gordon equation; see [50, 4, 5, 72, 73, 76] – no general framework could be established so far. Each and every equation had to be targeted carefully one at a time based on a sophisticated resonance analysis.…”

A Deuflhard-Type Exponential Integrator Fourier Pseudo-Spectral Method for the “Good” Boussinesq Equation

“…In more details, an unconditional full order convergence was shown for a second order operator splitting numerical scheme in the energy norm [33]: the H 2 × L 2 norm in z × z t . Nowadays, exponential time integrators have been widely applied for parabolic and hyperbolic equations [18,25,28,34]. Particularly, several efficient schemes were proposed for solving the GB equation [25] based on fourth-order exponential integrators of Runge-Kutta type.…”